In the Yorkshire accent the definite article is shortened to just t. E.g. 'I went on the bus' becomes something that sounds like 'I went ont bus'. How does one punctuate this? Is it 'I went on t'bus', which seems logical (as the apostrophe indicates missing letters) but does not reproduce the sound that one hears? Or could it be 'I went on't bus'?

In Lancashire, expressions like 'He worked at t'mill' and 'She went down t'Mill' were once common. They're usually spelt out this way. The pronunciation seems different from across the Pennines, though - more like 'He work təh mill' and 'She went dowən Mill' - no sign of the 't' from the 'the'.
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Edwin AshworthOct 4 '13 at 21:55

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@EdwinAshworth That comment should be an answer.
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MετάEdOct 4 '13 at 22:20

In Yorkshire, the sound does not in fact have a /t/ in it at all: it is usually a glottal stop, /ɔnʔbʊs/ (where /ʔ/ is a glottal stop) "on'bus". In rapid speech the /n/ often assimilates to the following labial /ɔmʔbʊs/ "om'bus", or the glottal stop disappears entirely (/ɔmbʊs/ "ombus").

English orthography has no standard way of writing this bit of regional English, but the most common spelling is "t'", so "I went on t'bus". As you say, this does not well represent the sounds, but that is true of much of English spelling.